The method of integration can be traced back to the Egyptians, in the Moscow Mathematical Papyrus circa 1800 BC, which gives the formula for finding the volume of a pyramidal frustrum.[1] Greek geometers are credited with a significant use of infinitesimals. Democritus is the first person recorded to consider seriously the division of objects into an infinite number of cross-sections, but his inability to rationalize discrete cross-sections with a cone's smooth slope prevented him from accepting the idea. At approximately the same time, Zeno of Elea discredited infinitesimals further by his articulation of the paradoxes which they create. Antiphon and later Eudoxus are generally credited with implementing the method of exhaustion, which made it possible to compute the area and volume of regions and solids by breaking them up into an infinite number of recognizable shapes. Archimedes developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat. (See Archimedes' Quadrature of the Parabola and Archimedes on Spheres & Cylinders.) It was not until the time of Newton that these methods were made obsolete. It should not be thought that infinitesimals were put on rigorous footing during this time, however. Only when it was supplemented by a proper geometric proof would Greek Mathematicians accept a proposition as true. (See Archimedes' use of infinitesimals.) The method of exhaustion was rediscovered in China by Liu Hui in the 3rd century AD, who used it to find the area of a circle. It was also used by Zu Chongzhi in the 5th century AD, who used it to find the volume of a sphere.[1]

Basic Limits

Example  จงหาค่า
  • แทนค่า x เข้าใกล้ 0 ทางลบ ในสมการ
  • งานต่อไปต้องแปลงสมการ เพื่อทำให้สามารถตัดอะไรซักอย่างก็ได้
  • ลองคูณบนคูณล่างด้วย
  • ตอนนี้ปัญหาอยู่ที่ ต้องถอด absolute ก่อนจึงจะสามารถตัดกับ x ข้างล่างนะ
  • สูตรการถอด absolute คือ
  • ข้อนี้โจทย์บอก x เข้าใกล้ 0 ทางลบ ดังนั้น
  • ตัด -x ทั้งบนและล่าง
  • แทนค่า x เข้าใกล้ 0ทางลบ ในสมการอีกครั้ง
  • อธิบายเพิ่ม
    ในห้ามติดลบ แต่เนื่องจากแทนค่า x ด้วย 0 ทางลบ ซึ่งเมื่อเจอกับลบอีกจึงทำให้ใน ยังเป็นบวกอยู่นั้นเอง
Other Questions
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Author: Surachai U.
Created Date: 2007-03-03 03:03:03
Last Update: 2007-06-01 23:33:08

The concept of a limit is fundamental to Calculus. In fact, Calculus without limits is like Romeo without Juliet. It is at the heart of so many Calculus concepts like the derivative, the integral, etc. So what is a limit? Maybe the best example to illustrate limits is through average and instantaneous speeds: Let us assume you are traveling from point A to point B while passing through point C. Then we know how to compute the average speed from A to B: it is simply the ratio between the distance from A to B and the time it takes to travel from A to B. Though we know how to compute the average speed this has no real physical meaning. Indeed, let us suppose that a policeman is standing at point C checking for speeders going through C. Then the policeman does not care about the average speed. He only cares about the speed that you see on the speedometer, the one that the car actually has when crossing C. That one is real.